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| Examples | |
| 2>
The unit circle S1 is a K(Z,1).
The infinite-dimensional complex projective space P∞(C) is a model of K(Z,2). This is one of the rare examples of classifying spaces admitting a manifold model, and is also the topological space the homotopy groups of which satisfy πi = 0 for i = 1 and i > 2, while π2 = Z. Its cohomology ring is Z[x], namely the free polynomial ring on a single 2-dimensional generator x ∈ H2. The generator can be represented in de Rham cohomology by the Fubini–Study 2-form. An application of K(Z,2) is described at Abstract nonsense.
The infinite-dimensional real projective space P∞(R) is a K(Z2,1).
The wedge sum of k unit circles is a K(G,1) for G the free group on k generators.
The complement to any knot in a 3-dimensional sphere S3 is of type K(G,1); this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos.[1]
Some further elementary examples can be constructed from these by using the obvious fact that the product K(G,n) × K(H,n) is K(G × H,n).
A K(G,n) can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group G, and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy.
[edit] Tags:Topological Space,Homotopy Groups,Homotopy Group,Edit,Unit Circle,Complex Projective Space,Manifold,De Rham Cohomology,Fubini–study,2-form,Abstract Nonsense,Real Projective Space,Wedge Sum,Free Group,Asphericity,Christos Papakyriakopoulos,Cw Complex,Wedge,Spheres, | |
| Properties of Eilenberg–MacLane spaces | |
| 2>
An important property of K(G,n) is that, for any abelian group G, and any CW-complex X, the set
[X, K(G,n)]
of homotopy classes of maps from X to K(G,n) is in natural bijection with the n-th singular cohomology group
Hn(X; G)
of the space X. Thus one says that the K(G,n) are representing spaces for cohomology with coefficients in G. Since , there is a distinguished element corresponding to the identity. The above bijection is given by pullback of that element — .
Another version of this result, due to Peter J. Huber, establishes a bijection with the n-th ÄŒech cohomology group when X is Hausdorff and paracompact and G is countable, or when X is Hausdorff, paracompact and compactly generated and G is arbitrary. A further result of Morita establishes a bijection with the n-th numerable ÄŒech cohomology group for an arbitrary topological space X and G an arbitrary abelian group.
Every CW-complex possesses a Postnikov tower, that is, it is homotopy equivalent to an iterated fibration with fibers the Eilenberg–MacLane spaces.
There is a method due to Jean-Pierre Serre which allows one, at least theoretically, to compute homotopy groups of spaces using a spectral sequence for special fibrations with Eilenberg–MacLane spaces for fibers.
The cohomology groups of Eilenberg–MacLane spaces can be used to classify all cohomology operations.
[edit] Tags:Cohomology Operations,Cw-complex,Singular Cohomology Group,ÄŒech Cohomology Group,Hausdorff,Paracompact,Compactly Generated,Numerable ÄŒech Cohomology Group,Postnikov Tower,Jean-pierre Serre,Spectral Sequence, | |
| See also | |
| 2>
Brown representability theorem, regarding representation spaces
Moore space, the homology analogue.
[edit] Tags:Brown Representability Theorem,Moore Space, | |
| Notes | |
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^ Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. MR13312.) In this context it is therefore conventional to write the name without a space.
^ (Papakyriakopoulos 1957)
[edit] Tags:Saunders Mac Lane, | |
| References | |
| 2>
S. Eilenberg, S. MacLane, Relations between homology and homotopy groups of spaces Ann. of Math. 46 (1945) pp. 480–509
S. Eilenberg, S. MacLane, Relations between homology and homotopy groups of spaces. II Ann. of Math. 51 (1950) pp. 514–533
Peter J. Huber (1961), Homotopical cohomology and Čech cohomology, Mathematische Annalen 144 , 73–76.
Morita Kiiti (1975). "Čech cohomology and covering dimension for topological spaces". Fundamenta Mathematicae 87: 31–52.
Papakyriakopoulos, C. D. (1957). "On Dehn's Lemma and the Asphericity of Knots". Proc. Nat. Acad. Sci. USA 43 (1): 169–172. doi:10.1073/pnas.43.1.169. PMC 528404. PMID 16589993. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=528404.
Papakyriakopoulos, C. D. (1957). "On Dehn's Lemma and the Asphericity of Knots". Ann. Math. 66 (1): 1–26. doi:10.2307/1970113. JSTOR 1970113.
Rudyak, Yu.B. (2001), "Eilenberg−MacLane space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=E/e035200
Retrieved from "http://en.wikipedia.org/w/index.php?title=Eilenberg%E2%80%93MacLane_space&oldid=448778682"
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Tags:Mathematics,Homotopy Theory,Algebraic Topology, | |
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